In a past life research was my thing. So time to put my money where my mouth is and do a bit of ye olde historical research.
As a former fire boat, there is bound to be some file on Beta III somewhere… and, low and behold, the National Archives say that the London Metropolitan Archives have a file. Excellent.
LCC/FB/GEN/3/44
LCC/FB/GEN/3/45 – this seems to be the main file.
They are literally a stones throw from my house.
Naomi – any other hints or tips about where to look?
betaiiifireboat 
You’ve definitely hit upon the right file at the London Metropolitan Archives – I doubt you’re going to find any other more specific records. You might however be interested in these clips from Pathe news which shows London firefloats in action between 1914-1932:
http://www.britishpathe.com/video/testing-fire-float/query/fire+boat
http://www.britishpathe.com/video/2-400-gallons-a-minute/query/fire+boat
http://www.britishpathe.com/video/fire-float-takes-a-ride/query/fire+boat
Possible that No. 2 (1925) is the Beta III? Perhaps the file will tell you more…
Good luck!
Ooooh. That is a distinct possibility. It does look pretty similar to Beta III. The only catch is that Beta III was built in 1926 and the video dates from 1925. Thank you!